Linear stationary dynamic systems

Linear, dynamic, time-invariant, lumped, continuous-time systems, can be modeled with (a set of) ordinary linear differential equations with constant coefficients. If such a system has only one input and one output, the input-output relation is given by a linear differential equation of the order \(n\) with fixed coefficients. The general form of such a differential equation will be:

(201)\[\sum_{i=0}^{i=n}a_{i}\frac{d^{i}y(t)}{dt^{i}}=\sum_{k=0}^{k=m}b_{k}\frac{d^{k}x(t)}{dt^{k}}, \label{differential-equation}\]

where \(x(t)\) is the input signal or excitation, \(y(t)\) the output signal or response, and \(a_{i}\) and \(b_{k\text{ }}\) the time-independent (fixed) real coefficients. Due to the physical limitation of speed, we have \(n>m.\)

The response of a linear dynamic system to an arbitrary input signal can be found by solving the above differential equation. Direct solution of a differential equation requires an exact time domain description of the excitation \(x(t),\) as well as the definition of \(n\) initial conditions. Such descriptions are not always available. However, since we are dealing with linear systems, the response of the system to an arbitrary signal, may also be obtained as a linear superposition of responses to elementary signals in which the arbitrary signal can be resolved. To this end, signals are decomposed into series of unit impulses \(\delta(t)\), imaginary exponentials \(\exp j\omega t\), or complex exponentials \(\exp st,\) where \(s=\sigma+j\omega\). In this section, we will discuss modeling techniques, based on such decompositions.

Time domain analysis

1. Solution with the aid of the unit impulse response \(h(t)=\mathcal{H} \{\delta(t)\},\)\ or the unit step response \(a(t)=\int_{-\infty}^{t} h(\tau)d\tau\).

   Since all time functions can be resolved into unit impulse or unit step functions, the response \(y(t)\) of a linear system to an arbitrary signal \(x(t)\) can be found as the sum of the responses to the unit impulse, or the unit step functions into which the signal is resolved. The operation for finding the time-domain response with the aid of resolution in unit impulses is called convolution:

   \[\begin{split}y(t) & =x(t)\ast h(t)=\int_{0}^{t}x(\tau)h(t-\tau)d\tau,\\ y(t) & =\int_{0}^{t}\dot{x}(\tau)a(t-\tau)d\tau=\dot{x}(t)\ast a(t).\end{split}\]

2. Analysis of the behavior using signals resolved in elementary exponentials.

   Since exponential functions retain their shape under the operation of differentiation and integration, differential equations can be transformed into algebraic equations if the excitation and response are written as exponential functions. The algebraic equation obtained in this way, can be solved analytically. For this reason, we often resolve signals in imaginary or complex exponentials, using the Fourier and Laplace transformation techniques, respectively. These techniques will be discussed below.

Frequency domain analysis

The resolution of an arbitrary time signal into imaginary exponentials is convenient both from the mathematical and practical point of view. Resolution of signals into elementary imaginary exponentials, allows us to use frequency domain descriptions for both information-carrying signals and test signals. Linear amplifiers can be characterized with the aid of sinusoidal test signals, and deviations from linear behavior can be observed as signal distortions at the amplifier’s load.

We will briefly demonstrate this technique for the differential equation given in (differential-equation). To this end, we resolve \(x(t)\) and \(y(t)\) into elementary imaginary exponentials. One element of the resolved excitation \(x(t)\) is then of the form \(X(j\omega)\exp j\omega t\) and the corresponding response is of the same shape and can be written as \(Y(j\omega)\exp j\omega t,\) in which \(X(j\omega)\) and \(Y(j\omega)\) are the complex amplitudes of the imaginary exponentials \(\exp j\omega t\) at the input and the output, respectively. The \(k-th\) derivative of an element of the resolved input signal is obtained as \((j\omega)^{k}X(j\omega)\exp j\omega t\) and the \(i-th\) derivative of its corresponding response as \((j\omega)^{i}Y(j\omega)\exp j\omega t\). All initial conditions are assumed to be zero. Substituting these exponentials into the differential equation yields

\[\sum_{i=0}^{i=n}a_{i}(j\omega)^{i}Y(j\omega)\exp j\omega t=\sum_{k=0} ^{k=m}b_{k}(j\omega)^{k}X(j\omega)\exp j\omega t.\]

We now obtain the transfer function \(H(j\omega)\) of the amplifier, which relates, for a given frequency \(\omega\), the complex amplitude of the response \(Y(j\omega)\) to that of the excitation \(X(j\omega)\):

\[H(j\omega)=\frac{Y(j\omega)}{X(j\omega)}=\frac{\sum_{k=0}^{k=m}b_{k} (j\omega)^{k}}{\sum_{i=0}^{i=n}a_{i}(j\omega)^{i}}.\]

Transfer function

The transfer function \(H(j\omega)\) of a linear fixed dynamic system is the Fourier transform of the system’s unit impulse response:

\[H(j\omega)=\mathcal{F}\{h(t)\}.\]

Information-carrying signals are often characterized by their power spectral density \(S(\omega)\), which is a statistical quantity defined as the average signal power in [W/Hz] at a certain frequency, over a bandwidth of one Hz. If the input signal \(x(t)\) of a system with a transfer function \(H(j\omega)\), has a power spectral density of \(S_{x}(\omega)\), the power spectral density \(S_{y}(\omega)\) of the response signal \(y(t)\) can be obtained as:

\[S_{y}(\omega)=S_{x}(\omega)\left\vert H(j\omega)\right\vert ^{2}.\]

Bode plots

A graphical representation of the transfer function can be given by plotting the magnitude \(\left\vert H(j\omega)\right\vert \) and the argument \(\arg\{H(j\omega)\}\) of the transfer function \(H(j\omega)\) as a function of \(\omega.\) These so-called Bode plots show the magnitude \(20\log{} _{10}\left\vert H(j\omega)\right\vert \) in dB and the argument \(\arg \{H(j\omega)\}\) in degrees or radians, both on a linear scale. The angular frequency \(\omega\) or the frequency \(f\) is usually plotted on a logarithmic scale.

Complex frequency domain analysis

The system function \(H(s)\) is obtained in a similar way as the transfer function \(H(j\omega)\); it relates the complex amplitudes \(Y(s)\) of the complex exponentials of the resolved output signal to\ those of the resolved input signal. \(H(s)\) is the Laplace transform of the amplifier’s unit impulse response:

(202)\[H(s)=\frac{Y(s)}{X(s)}=\frac{\sum_{k=0}^{k=m}b_{k}s^{k}}{\sum_{i=0}^{i=n} a_{i}s^{i}}=\mathcal{L}\{h(t)\}. \label{system-function}\]

The Laplace transform is also defined for signals that have no power limitation. These (theoretical) signals arise in linear(ized) unstable dynamic systems. We will show that the stability of linear(ized) dynamic systems can be investigated through evaluation of the system function.

The numerator of the system function (see expression system-function) is a polynomial of degree \(m.\) The \(m\) roots of the numerator are called the zeros of the system functions. The \(n\) roots of the denominator are called the poles of the system function. Any system function of a system that does not incorporate delay lines can be written in terms of these poles and zeros:

\[H(s)=\frac{b_{m}}{a_{n}}\frac{\prod\limits_{k=1}^{m}(s-z_{k})}{\prod \limits_{i=1}^{n}(s-p_{i})},\]

where \(z_{k}\) is a zero of \(H(s)\) and \(p_{i}\) is a pole of \(H(s)\).

If there are no poles or zeros at \(s=0\), this may also be written as

(203)\[H(s)=\frac{b_{0}}{a_{0}}\frac{\prod\limits_{k=1}^{m}(1-\frac{s}{z_{k}})}{\prod\limits_{i=1}^{n}(1-\frac{s}{p_{i}})}. \label{ex-systemfunctionDC}\]

The system function is completely described by means of its poles and zeros and the coefficients \(a_{0}\) and \(b_{0}\) or \(a_{n}\) and \(b_{m}\). The factor \(\frac{b_{0}}{a_{0}}\) is the zero frequency transfer or the DC transfer of the system.

Pole-zero pattern

The graphical representation of the poles and zeros in the complex plane is referred to as the pole-zero pattern of \(H(s)\). A typical pole-zero pattern of a system function is depicted in Fig. 530.

Fig. 530 Example of a pole-zero pattern.

Poles and zeros always appear as single real or pairs of complex conjugates; this is because the coefficients of the differential equations are real.

Not all system functions can be represented by a pole-zero pattern. A system with a time delay \(\tau\) has a system function like: \(H(s)=\exp\left( -s\tau\right) ,\) which cannot be expressed in terms of poles and zeros.

Pole-zero pattern and Bode plots

The relation between the Bode plots and the pole-zero pattern can be found by substituting \(s=j\omega\) in the expression for the system function \(H(s)\). The magnitude of the transfer function \(\left\vert H(j\omega)\right\vert \) can then be evaluated from

\[\left\vert H(j\omega)\right\vert =\frac{b_{m}}{a_{n}}\frac{\prod \limits_{k=0}^{k=m}\left\vert j\omega-z_{k}\right\vert }{\prod\limits_{i=0} ^{i=n}\left\vert j\omega-p_{i}\right\vert }.\]

Fig. 531 Relation between pole-zero pattern and Bode plots.

The magnitude of the transfer function is thus proportional to the quotient of the product of the magnitudes of the terms of the numerator and the product of the magnitudes of the denominator. The magnitude of a single factor \(\left\vert j\omega-z_{k}\right\vert \) at an angular frequency \(\omega_{1}\) equals the distance between the location of the zero \(z_{k}\) and the position \(j\omega=j\omega_{1}\) on the imaginary axis. The magnitude \(\left\vert j\omega-p_{i}\right\vert \) can be found in a similar way. This is shown in Fig. 531.

The argument of the transfer function can be evaluated as:

\[\arg\{H(j\omega)\}=\arg b_{m}-\arg a_{n}+\sum\limits_{k=0}^{k=m}\arg (j\omega-z_{k})-\sum\limits_{i=0}^{i=n}\arg(j\omega-p_{i}).\]

The arguments of \(a_{n}\) and \(b_{m}\) are either \(0\) or \(\pi\) rad, depending on their sign. The argument of a single term \((j\omega-z_{k})\) at an angular frequency \(\omega_{1}\) is found as the angle formed by the vector that connects the zero \(z_{k}\) with the point \(j\omega=j\omega_{1}\) on the imaginary axis and the positive real axis. This relation is shown in Fig. 531.

Minimum phase system

A system is called a minimum phase system if the inverse of \(H(s)\) is stable and causal. This implies that \(H(s)\) has no zeros in the right half plane. The magnitude and the phase characteristic of a minimum phase system are related by the Hilbert Transform:

\[\arg\left\{ H\left( j\omega\right) \right\} =-\mathcal{H}_{\mathcal{I} }\left\{ \ln\left\vert H\left( j\omega\right) \right\vert \right\} ,\]

in which the Hilbert transform operator \(\mathcal{H}_{\mathcal{I}}\) is defined as:

\[\mathcal{H}_{\mathcal{I}}\left\{ x\left( t\right) \right\} =\frac{1}{\pi }\int_{-\infty}^{\infty}\frac{x(\tau)}{t-\tau}d\tau.\]

In the frequency domain, this is equivalent to

\[\mathcal{H}_{\mathcal{I}}\left\{ X\left( j\omega\right) \right\} =-j\operatorname*{sgn}\omega~X\left( j\omega\right) .\]

A minimum phase system has the smallest possible delay of all linear time-invariant dynamic systems that have equal magnitude characteristics.

Time domain analysis using the Laplace transform

Time domain analysis, using signals resolved in elementary exponentials, is performed with the aid of the Laplace transform. The procedure for the determination of the time domain response to a known time function can be presented as

\[y(t)=\mathcal{L}^{-1}\{Y(s)\}=\mathcal{L}^{-1}\{H(s)X(s)\}=\mathcal{L} ^{-1}[H(s)\mathcal{L}\{x(t)\}].\]

For determination of the Laplace transform of a time function, we use tables of the Laplace pairs and the properties of the Laplace transform.

Stability

The system’s unit impulse response \(h(t)\) can be found from the inverse Laplace transform of its system function \(H(s).\) The unit impulse response of a system with more poles than zeros can be written as a sum of exponentials with \(p_{k}t\) as its argument, in which \(p_{k}\) is a pole of the system function. A general expression for the impulse response of a system with more poles than zeros is:

(204)\[h(t)=\sum_{i=1}^{n}\sum_{j=0}^{\ell-1}A_{i,j}t^{j}\exp p_{i}t \label{eq-ir}\]

in which \(\ell\) is the number of occurrences for the pole \(p_{i}\). The coefficients \(A_{i,j}\) depend on the poles and the zeros. Expression (204) clearly shows that the impulse response is bounded if all poles of the system function have a negative real part. As a consequence, if all poles have a negative real part, the response to any bounded signal is bounded and the system is stable.

Definition

Definition: A system is said to be stable, if the poles of its system function \(H(s)\) are all located in the left half of the complex \(s\)-plane.